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<li class="toctree-l3"><a class="reference internal" href="#import">Import</a></li>
<li class="toctree-l3"><a class="reference internal" href="#model-and-data">Model and Data</a></li>
<li class="toctree-l3"><a class="reference internal" href="#filter-design">Filter Design</a><ul>
<li class="toctree-l4"><a class="reference internal" href="#filter-initialization">Filter Initialization</a></li>
<li class="toctree-l4"><a class="reference internal" href="#filtering">Filtering</a></li>
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<li class="toctree-l3"><a class="reference internal" href="#results">Results</a></li>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-wifibot-py"><span class="std std-ref">here</span></a> to download the full example code</p>
</div>
<div class="sphx-glr-example-title section" id="d-robot-localization-on-real-data">
<span id="sphx-glr-auto-examples-wifibot-py"></span><h1>2D Robot Localization on Real Data<a class="headerlink" href="#d-robot-localization-on-real-data" title="Permalink to this headline">¶</a></h1>
<p>Goals of this script:</p>
<ul class="simple">
<li><p>apply the UKF for the 2D robot localization example on real data.</p></li>
</ul>
<p><em>We assume the reader is already familiar with the considered problem described
in the tutorial.</em></p>
<p>We address the same problem as described in the tutorial on our own data.</p>
<div class="section" id="import">
<h2>Import<a class="headerlink" href="#import" title="Permalink to this headline">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ukfm</span> <span class="k">import</span> <span class="n">LOCALIZATION</span> <span class="k">as</span> <span class="n">MODEL</span>
<span class="kn">import</span> <span class="nn">ukfm</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib</span>
<span class="n">ukfm</span><span class="o">.</span><span class="n">utils</span><span class="o">.</span><span class="n">set_matplotlib_config</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="section" id="model-and-data">
<h2>Model and Data<a class="headerlink" href="#model-and-data" title="Permalink to this headline">¶</a></h2>
<p>This script uses the <a class="reference internal" href="../model.html#ukfm.LOCALIZATION" title="ukfm.LOCALIZATION"><code class="xref py py-meth docutils literal notranslate"><span class="pre">LOCALIZATION()</span></code></a> model.</p>
<p>Instead of creating data, we load recorded data. We have recorded five
sequences (sequence 2 and 3 are the more interesting).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># sequence number</span>
<span class="n">n_sequence</span> <span class="o">=</span> <span class="mi">3</span>
<span class="c1"># GPS frequency (Hz)</span>
<span class="n">gps_freq</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c1"># GPS noise standard deviation (m)</span>
<span class="n">gps_std</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="c1"># load data</span>
<span class="n">states</span><span class="p">,</span> <span class="n">omegas</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">one_hot_ys</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">MODEL</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">n_sequence</span><span class="p">,</span> <span class="n">gps_freq</span><span class="p">,</span> <span class="n">gps_std</span><span class="p">)</span>
</pre></div>
</div>
<p>Data has been obtained in an experiment conducted at the Centre for Robotics,
MINES ParisTech. We used a so-called Wifibot, which is a small wheeled robot
equipped with independent odometers on the left and right wheels, see figure.
A set of seven highly precise cameras, the OptiTrack motion capture system,
provide the reference trajectory (ground truth) with sub-millimeter precision
at a rate of 120 Hz.</p>
<div class="figure align-center" id="id1">
<a class="reference internal image-reference" href="../_images/robot.jpg"><img alt="robot picture." src="../_images/robot.jpg" style="width: 420.40000000000003px; height: 509.40000000000003px;" /></a>
<p class="caption"><span class="caption-text">Testing arena with Wifibot robot in the foreground of the picture. We can
also see two of the seven Optitrack cameras in the background.</span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
</div>
<p>We define noise odometry standard deviation for the filter.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">odo_std</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.15</span><span class="p">,</span>   <span class="c1"># longitudinal speed</span>
                    <span class="mf">0.05</span><span class="p">,</span>   <span class="c1"># transverse shift speed</span>
                    <span class="mf">0.15</span><span class="p">])</span>  <span class="c1"># differential odometry</span>
</pre></div>
</div>
</div>
<div class="section" id="filter-design">
<h2>Filter Design<a class="headerlink" href="#filter-design" title="Permalink to this headline">¶</a></h2>
<p>We embed here the state in <span class="math notranslate nohighlight">\(SE(2)\)</span> with left multiplication, i.e.</p>
<blockquote>
<div><ul class="simple">
<li><p>the retraction <span class="math notranslate nohighlight">\(\varphi(.,.)\)</span> is the <span class="math notranslate nohighlight">\(SE(2)\)</span> exponential, where
the state multiplies on the left the uncertainty <span class="math notranslate nohighlight">\(\boldsymbol{\xi}\)</span>.</p></li>
</ul>
</div></blockquote>
<ul class="simple">
<li><p>the inverse retraction <span class="math notranslate nohighlight">\(\varphi^{-1}_.(.)\)</span> is the <span class="math notranslate nohighlight">\(SE(2)\)</span>
logarithm.</p></li>
</ul>
<p>We define the filter parameters based on the model parameters.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># propagation noise covariance matrix</span>
<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">odo_std</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="c1"># measurement noise covariance matrix</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">gps_std</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># sigma point parameters</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1e-3</span><span class="p">,</span> <span class="mf">1e-3</span><span class="p">,</span> <span class="mf">1e-3</span><span class="p">])</span>
</pre></div>
</div>
<div class="section" id="filter-initialization">
<h3>Filter Initialization<a class="headerlink" href="#filter-initialization" title="Permalink to this headline">¶</a></h3>
<p>We initialize the filter with the true state plus an initial heading error of
30°, and set corresponding initial covariance matrices.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># &quot;add&quot; orientation error to the initial state</span>
<span class="n">SO2</span> <span class="o">=</span> <span class="n">ukfm</span><span class="o">.</span><span class="n">SO2</span>
<span class="n">state0</span> <span class="o">=</span> <span class="n">MODEL</span><span class="o">.</span><span class="n">STATE</span><span class="p">(</span><span class="n">Rot</span><span class="o">=</span><span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">Rot</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">SO2</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">30</span><span class="o">/</span><span class="mi">180</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)),</span>
                     <span class="n">p</span><span class="o">=</span><span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
<span class="c1"># initial state uncertainty covariance matrix</span>
<span class="n">P0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="c1"># The state is not perfectly initialized</span>
<span class="n">P0</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">30</span><span class="o">/</span><span class="mi">180</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
</pre></div>
</div>
<p>We define the filter as an instance of the <code class="docutils literal notranslate"><span class="pre">UKF</span></code> class.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">ukf</span> <span class="o">=</span> <span class="n">ukfm</span><span class="o">.</span><span class="n">UKF</span><span class="p">(</span><span class="n">state0</span><span class="o">=</span><span class="n">state0</span><span class="p">,</span>               <span class="c1"># initial state</span>
               <span class="n">P0</span><span class="o">=</span><span class="n">P0</span><span class="p">,</span>                       <span class="c1"># initial covariance</span>
               <span class="n">f</span><span class="o">=</span><span class="n">MODEL</span><span class="o">.</span><span class="n">f</span><span class="p">,</span>                   <span class="c1"># propagation model</span>
               <span class="n">h</span><span class="o">=</span><span class="n">MODEL</span><span class="o">.</span><span class="n">h</span><span class="p">,</span>                   <span class="c1"># observation model</span>
               <span class="n">Q</span><span class="o">=</span><span class="n">Q</span><span class="p">,</span>                         <span class="c1"># process noise covariance</span>
               <span class="n">R</span><span class="o">=</span><span class="n">R</span><span class="p">,</span>                         <span class="c1"># observation noise covariance</span>
               <span class="n">phi</span><span class="o">=</span><span class="n">MODEL</span><span class="o">.</span><span class="n">left_phi</span><span class="p">,</span>          <span class="c1"># retraction function</span>
               <span class="n">phi_inv</span><span class="o">=</span><span class="n">MODEL</span><span class="o">.</span><span class="n">left_phi_inv</span><span class="p">,</span>  <span class="c1"># inverse retraction function</span>
               <span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span>                  <span class="c1"># sigma point parameters</span>
               <span class="p">)</span>
</pre></div>
</div>
<p>Before launching the filter, we set a list for recording estimates along the
full trajectory and a 3D array to record covariance estimates.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">ukf_states</span> <span class="o">=</span> <span class="p">[</span><span class="n">ukf</span><span class="o">.</span><span class="n">state</span><span class="p">]</span>
<span class="n">ukf_Ps</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="n">ukf_Ps</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">ukf</span><span class="o">.</span><span class="n">P</span>
</pre></div>
</div>
</div>
<div class="section" id="filtering">
<h3>Filtering<a class="headerlink" href="#filtering" title="Permalink to this headline">¶</a></h3>
<p>The UKF proceeds as a standard Kalman filter with a for loop.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># measurement iteration number (first measurement is for n == 0)</span>
<span class="n">k</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
    <span class="c1"># propagation</span>
    <span class="n">dt</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">-</span> <span class="n">t</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
    <span class="n">ukf</span><span class="o">.</span><span class="n">propagation</span><span class="p">(</span><span class="n">omegas</span><span class="p">[</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">dt</span><span class="p">)</span>
    <span class="c1"># update only if a measurement is received</span>
    <span class="k">if</span> <span class="n">one_hot_ys</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
        <span class="n">ukf</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">ys</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
        <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
    <span class="c1"># save estimates</span>
    <span class="n">ukf_states</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ukf</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
    <span class="n">ukf_Ps</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">ukf</span><span class="o">.</span><span class="n">P</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="results">
<h2>Results<a class="headerlink" href="#results" title="Permalink to this headline">¶</a></h2>
<p>We plot the trajectory, the measurements and the estimated trajectory. We then
plot the position and orientation error with 95% (<span class="math notranslate nohighlight">\(3\sigma\)</span>) confident
interval.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">MODEL</span><span class="o">.</span><span class="n">plot_wifibot</span><span class="p">(</span><span class="n">ukf_states</span><span class="p">,</span> <span class="n">ukf_Ps</span><span class="p">,</span> <span class="n">states</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
</pre></div>
</div>
<ul class="sphx-glr-horizontal">
<li><img alt="../_images/sphx_glr_wifibot_001.png" class="sphx-glr-multi-img" src="../_images/sphx_glr_wifibot_001.png" />
</li>
<li><img alt="../_images/sphx_glr_wifibot_002.png" class="sphx-glr-multi-img" src="../_images/sphx_glr_wifibot_002.png" />
</li>
<li><img alt="../_images/sphx_glr_wifibot_003.png" class="sphx-glr-multi-img" src="../_images/sphx_glr_wifibot_003.png" />
</li>
</ul>
<p>All results are coherent. This is convincing as the initial heading error is
relatively high.</p>
</div>
<div class="section" id="conclusion">
<h2>Conclusion<a class="headerlink" href="#conclusion" title="Permalink to this headline">¶</a></h2>
<p>This script applies the UKF for localizing a robot on real data. The filter
works well on this localization problem on real data, with moderate
initial heading error.</p>
<p>You can now:</p>
<ul class="simple">
<li><p>test the UKF on different sequences.</p></li>
<li><p>address the UKF for the same problem with range and bearing measurements of
known landmarks.</p></li>
</ul>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes  7.655 seconds)</p>
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